She has always been a divisive figure. Ever since the world agreed that Yoko Ono was responsible for “breaking up The Beatles”, she has been the target of a Molotov cocktail of misogyny, jealousy and racism. Added to which, her singing style can be challenging.

When the album Double Fantasy was released in November 1980, John Lennon and Yoko taking alternate tracks, Ono-phobia guaranteed that reviews were mixed. But have we, at last, got over our prejudice? As she presents the album in its entirety as the finale of her warmly received Meltdown Festival, it would appear that we have.

The emotional big guns are wheeled out early, as we’re shown a clip of the couple crossing New York’s Central Park. Lennon is accosted by a dizzy autograph hunter, and an effusive dork grabs his hand through a fence: “When are The Beatles getting back together? I love your blue album ...” As the pair stroll off through autumn leaves, John’s words – “Well, here we are again, just two ordinary people walking through a park” – are left to hang in the air as we digest what Lennon’s approachability would mean only a month later.

Upon which The Plastic Ono Band, who include John and Yoko’s son Sean (looking the spit of his father) and original sideman Earl Slick, strike up the Presleyesque single “(Just Like) Starting Over”, with singer-songwriter Pete Molinari doffing a respectful trilby towards Yoko in the royal box.

Patti Smith is the first to break omerta, giving a dedicatory speech before “Beautiful Boy”, which contains Lennon’s famous line “Life is what happens to you while you’re busy making other plans”. She’s accompanied on harp by Patrick Wolf, who then delivers a tremendous “Watching the Wheels” alone. To emphasise the female-first flavour of the whole festival, a quartet of Peaches, Camille, Bishi and Lene gleefully tackle the Weimar romp of “Yes, I’m Your Angel”. There’s another Lennon home movie as John, poorly lit by a table lamp, sings “Dear Yoko” into a camcorder. Then it’s the turn of Boy George, fresh from his own Meltdown show and looking lean and gorgeous, and sounding it too on “Hard Times Are Over”.

After an encore of “Walking on Thin Ice”, Yoko’s superb death-disco single from 1981 imperiously performed by Siouxsie Sioux, there’s suddenly a shrill “ay, ay, ay, ay” from stage-right, and a tiny octogenarian Japanese lady walks out. The sometime hate figure is greeted with a wave of purest adoration.

A similar ovation greets another octogenarian in the same room three days later. If The Beatles made the Sixties rock, Burt Bacharach made them swing. At 85, sporting a smart suit and comfy trainers, the silver fox leads his band through a well-drilled two-hour segue of his mind-boggling repertoire, interspersed with memories of his London days spent writing timeless hits between boozing in the Grenadier pub in Belgravia.

He pays tribute to his co-writer, the late Hal David. The partnership disintegrated in 1972, while working on the score for a flop remake of the film Lost Horizon, when Burt decided he deserved a 3/2 split of royalties instead of 50/50, which Hal refused.

But what a body of work they’d accrued already: “Walk on By”, “The Look of Love” and “Close to You” are among the duo’s immortal compositions. But Burt doesn’t brush over his dafter moments tonight, like “What’s New, Pussycat?”, “Magic Moments” and the theme from The Blob, Steve McQueen’s first movie. “His career survived that, somehow. And so did mine ...”

It’s a joy to see someone still deriving such joy from keying those chord changes into a grand piano that’s longer than a London taxi. Some of the classics are spoiled by singer John Pagano, whose overwrought emoting, with a constipated grunt at the start of every line, feels better suited to American Idol than Bacharach-David. But when Burt himself takes lead vocal, it brings the house down.

He always imagined he’d be a classical composer, he tells us, and when you strip away the schmaltz, you can hear that. Bacharach’s songs possess a Bach-like logic and simplicity, which is the key to their adaptability (and the reason others, have been able to improvise on them so fruitfully). “They sound simple,” he confides. “That’s the trick.”

CRITIC'S CHOICE

The Manchester International Festival opens with Massive Attack vs Adam Curtis, a collaboration between the band, with Martina Topley-Bird and the documentary maker, at Mayfield Depot, Manchester (Thu to Sun, then 10 to 13 July). Power pop legends The Flamin’ Groovies, fresh from supporting Bruce Springsteen, shake some action at the Scala, London (Tue).